Polymer-Residue Accessibility Shapes Sequence Dependence of Critical Temperatures for Phase Separation

Abstract

Biological polymers, such as intrinsically disordered proteins, play a central role in cellular biology, including mediating phase separation and controlling activity of biological condensates. The physical properties and functions of biopolymers are determined by their residue sequence. Recently, significant computational and theoretical efforts have been devoted to characterizing the combinatorially complex sequence dependence of biopolymer phase diagrams. Here, we quantitatively show that monomer accessibility is central to determining the strength of pair interactions. We formulate an analytical perturbative approach, phenomenologically precluding two polymers' centers of mass from overlapping within a correlation hole. This theory yields the correction to the strength of mean-field interactions in terms of a residue-accessibility parameter (RAP), which accounts for the limited availability of inner monomers to interactions. Despite the simplicity of the approach, RAP rationalizes the variations in critical temperatures found in extensive Monte-Carlo simulations for thousands of two-letter polymer solutions of varying length and sequence. RAP may thus be effective for deciphering the polymer-sequence dependence of phase diagrams given any polymer length, set of monomer types, and polymer mixtures.

Figures (4)

• Figure 1: Modeling the effect of residue accessibility on the critical temperature. (a) Overlap between the centers of mass (CMs) of two polymers is penalized when they approach within each other’s correlation hole. Each polymer consists of residues indexed $i=1,\ldots,N$ in a specified sequence (e.g., blue beads represent attractive monomers, and beige represents inert monomers). The Kuhn length is denoted $b$. The spatial position of the $i$th monomer of the $n$th polymer is denoted $\mathbf{r}_{n,i}$. The $n$th polymer's CM, $\mathbf{r}_n^\mathrm{cm}$, is determined by the mean of the $N$ monomer positions. Since polymer CMs are limited to beyond the correlation hole, inner monomers are statistically less accessible for interactions than peripheral ones, as indicated by the bead color's shade. (b) When more attractive monomers are placed at the polymer periphery, they are more accessible for interactions with other polymers under the correlation-hole penalty. Stronger pair-interactions increase the critical temperature for phase separation. Illustrated are three polymer sequences with ranked critical temperatures. (c) Mean-squared distance (MSD) of a monomer from its host polymer's CM for a Gaussian chain versus monomer position, Eq. \ref{['eq:MSD']}, determining monomer's accessibility. (d) Schematic phase diagram for phase separation. The shaded regions correspond to the phase-coexistence regime for each of two polymers solutions; the complementary regions indicate a single-phase regime. The region bounded by a higher critical temperature corresponds to a polymer with more accessible attractive monomers.
• Figure 2: Simulated critical temperature $T_\mathrm{c}$ (rescaled according to Eq. \ref{['eq:Tctilde']}) versus the fraction of $\mathrm{A}$-type monomers along a chain, $f_\mathrm{A}$. Were the mean-field prediction, Eq. \ref{['eq:chi_id']}, accurate, we would have found $k_\mathrm{B}T_\mathrm{c}(1+1/N^{1/2})^2/(z\bar{\varepsilon})=1$ for the entire data set. Clearly, $f_\mathrm{A}$ is insufficient for delineating the sequence dependence of critical temperature.
• Figure 3: The residue accessibility arises from the correlation hole effect and is correlated with critical temperature. (a) Total correlation function, $h_{\mathrm{cm}}$ among polymer centers of mass for different homopolymer lengths $N$, computed in the dense phase at $0.7T_\mathrm{c}$, as a function of radial coordinate $r$. The collapse of all curves upon rescaling by $N^{1/2}$ supports our assumption for the universality of a correlation hole. (b) The residue-accessibility parameter (RAP) $P$ and the critical temperature $T_\mathrm{c}$ for $\mathrm{A}\mathrm{B}_7$ chains. ($P$ is given in Eq. \ref{['eq:RAP']}, using $\epsilon_{\mathrm{A}\mathrm{A}}=7/8$ and $\epsilon_{\mathrm{A}\mathrm{B}}=\epsilon_{\mathrm{B}\mathrm{A}}=\epsilon_{\mathrm{B}\mathrm{B}}=1/8$.) The larger is RAP, the less accessible is the attractive residue, and the lower is the critical temperature.
• Figure 4: Data collapse of Monte-Carlo simulation results for 2408 unique polymer sequences of various lengths (as indicated) $N$ and solvent qualities $c$. Critical temperature $T_\mathrm{c}$ (rescaled according to Eq. \ref{['eq:Tctilde']}) versus RAP (normalized by the constant $P_0\simeq32.76$ --- the RAP of a homopolymer, $\epsilon_{ij}=1$, of length $N\to\infty$). The intercept for $P=0$ is fixed at $k_\mathrm{B}T_\mathrm{c}(1+1/N^{1/2})^2/(z\bar{\varepsilon})=1$. The slope, corresponding to the ratio between the correlation hole and Gaussian-chain radius of gyration, is the only fitting parameter, and is determined by the average of the homopolymer results (outlined in red).